ON THE THEORY OF NUMBERS. 315 



to the demonstration, but here, as in some other instances, Euler is satisfied 

 with an induction which does not amount to a rigorous proof. The first ad- 

 missible proof of the resolubility of the equation was given by Lagrange in 

 the Melanges de la Societe de Turin, vol. iv. p. 41. He there shows that in 

 the development of VD, we shall obtain an infinite number of solutions of 

 some equation of the form T'— DU^=A, and that, by multiplying together 

 a sufficient number of these equations, we can deduce solutions of the equa- 

 tion T^— DU^=1. But the simpler demonstration of its solubility, which is 

 now to be found in most books on algebra, and which depends on the com- 

 pletion of the theory (left unfinished by Euler) of the development of a 

 quadratic surd in a continued fraction, was first given by Lagrange in the 

 Hist, de I'Academie de Berlin for 1767 and 1768, vol. xxiii. p. 272, vol. xxiv. 

 p. 236; and, in a simpler form, in the Additions to Euler's Algebra, art. 37. 

 Lastly, Gauss, who in the Disq. Arith. avoids the use of continued fractions, 

 has shown that if we form by the method which he indicates, the period of 

 any quadratic form of det. D, we may infer the complete solution of the 

 Equation T"— DU^=1, or =4', from the automorphics of any reduced form, 

 according as the form is properly or improperly primitive. (Disq. Arith. art. 

 198-202.) 



To express conveniently the principal theorems relating to these equations, 

 we employ the following notatioa *. The numerator of the continued 

 fraction 



is called the cumulant of the numbers q^, q^, • • • q„, and is represented by 

 the symbol {q^, q^, q^,. . .q„) ; the denominator is evidently the cumulant 

 ($'2' ^a' • • ^n)- Accents are sometimes employed to indicate that the first or 

 last quotient of a cumulant is to be omitted ; thus '(y^, q„, q^, . ■ . q ) 

 = (q^, q„... qj, (q^, q„ q^, . . . qj=(q„q„ q^, . . q„_^ ), '(q^, q^, . . qj 

 —(92' 9'3' • • 9n-i)' ^ periodic cumulant is represented by the notation 

 (^ij ?2. • • q,iX^ the suffix indicating the number of times which the period 

 is repeated, and a point being placed over the first and last quotients of the 

 period. In what follows m represents 1 or 2, according as we are considerinc 

 the equation T^— DU-=1, or =4. ° 



(i.) If /ij, n^, . . . /i2jt t>e the period of integral quotients in the develop- 

 ment of either root of a quadratic equation of determinant D, which we 

 suppose properly or improperly primitive according as m=l, or jm = 2, the 

 positive numbers T, and U^. which satisfy the equation T^— DU^=»i^ are ail 

 contained in the formulae 



If4(A,+Aj, 

 U,_ B, A.-A, r. 



■ X 



where 



m -a, -2/3, 



A / * * 



^x={.t^V A'2» • • /^2t)x» ^x=(^l» M2> • •• h2k)x> 



* This notation is due to Euler (see Nov, Comm. Pet. vol. ix. p. 53, and the memoir 

 already cited, " De usu novi algorithmi in Problemate Pelliano solvendo." Comment. Arith. 

 vol. i. p. 316). The convenient term " cumulant " has been introduced by Professor Sylvester 

 (Phil. Trans, vol. cxliii. p. 474), who has also suggested the use of accents to indicate the 

 omission of initial or final quotients. 



